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ON  THE  COLOUIIS  OF  THICK  PLATES.                     171
suppose the pupil a point, and reduce each pencil entering it to a single ray, which forms the axis of the pencil. Let E be the eye, or rather the centre of the pupil, h its perpendicular distance from the mirror, and suppose the axis of z to pass through E. The ray by which a portion of a band is seen as if at M cuts the mirror in a point whose co-ordinates x, y are equal to a', V altered in the ratio of h to h  c, so that
*'    -(L~h)y-
Substituting in (15), we get
from which it may be observed that cf has disappeared, as evidently ought to be the case. The expression (16) might have been at once deduced from (15) by putting the co-ordinates of the eye in place of a, b', c. The reason of this is evident, because the retardation is constant for the same ray, and a ray may be defined by the positions of any two points through which it passes. We may therefore employ the points E and P, instead of M and P, to define the ray, and may therefore at once substitute the coordinates of E for those of M in the expression for the retardation.
10. To determine the forms, &c. of the bands, nothing more will be requisite than to discuss the formula (16). As however this formula was obtained as a particular case from a very general, and consequently rather complicated investigation, in which the curvatures of the surfaces were supposed to have any values, and as the bands to which it relates are of great interest, the reader may be pleased to see a special investigation of the formula for                     \'i
the case of a plane mirror.                                                                                  ;|
Retaining the same notation as before, except where the con-                      |fy
trary is specified, let jL0, Etl be the feet of the perpendiculars let                          .jj
fall  from  Z, E on  the  plane of the  dimmed surface, and let                           j-!;
i0P = 6', E^P  u.    Let Jll be the retardation of a ray regularly                      !;;'!
refracted  and  reflected, scattered  at  emergence  at  P,  and  so                     ||l
reaching E\ R2 the retardation of a ray reaching E after having been scattered at P on entering into the glass, and let R1  R2 = R. Let LSTPE be the course of the first ray, which emanates from